Gamma frailty hazard ratio ($r(t)=a\frac{1+v{H}_0(t)}{1+va{H}_0(t)}$)

Let a frailty random variable have a gamma distribution with mean one and variance $v>0$. Its shape and rate are both $1/v$. For conditional hazard function $U{a}^z{h}_0(t)$, $z\in \{0,1\}$, averaging the conditional survivor function gives $(1+v{a}^z{H}_0(t){)}^{-1/v}$. Differentiating its logarithm yields the population hazard function $a^z{h}_0(t)/(1+v{a}^z{H}_0(t))$ and hence the displayed hazard ratio. Writing $x=v{H}_0(t)$ gives $r-1=(a-1)/(1+ax)$. Therefore the population hazard ratio moves monotonically from $a$ towards one as exposure $x$ increases, even though the individual conditional hazard ratio remains $a$. The limit one requires $H_0(t)\to \infty$ for fixed positive $v$, or $v\to \infty$ for fixed positive $H_0(t)$; it is not a consequence of elapsed time alone. This is survival selection through preferential removal of larger frailties.